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Номер 49, страница 91 - гдз по физике 10-11 класс задачник Гельфгат, Генденштейн
Авторы: Гельфгат И. М., Генденштейн Л. Э., Кирик Л. А.
Тип: Задачник
Издательство: Илекса
Год издания: 2008 - 2026
Уровень обучения: профильный
Цвет обложки: красный лупа, парень едет на велосипеде
ISBN: 978-5-89237-252-7
Популярные ГДЗ в 10 классе
Механика. Механические колебания и волны. 12. Механические колебания. Олимпиадные задачи - номер 49, страница 91.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319526 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "91" "field_page_end" => "92" "field_url" => "/10-klass/fizika/gelfgat-zadachnik/olimp-49" "field_display_title" => "49" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1071} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1299 #items: array:4 [ 0 => App\Models\Branch {#1072 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Book {#1076 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1075} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1108 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319303 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Book {#1111 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:1 [ 0 => App\Models\Branch {#1143 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …2} ] #original: array:24 [ "id" => 1319266 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механика" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319303 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109} "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1110} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1147 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319304 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "12. Механические колебания" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Book {#1150 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Branch {#1182 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319303 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1184 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #original: array:24 [ "id" => 1319303 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Механические колебания и волны" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1184 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1319304 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "12. Механические колебания" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1181} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1221 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1319306 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Олимпиадные задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1222 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "91" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Book {#1224 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1230 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1232 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1237 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1245 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_under_the_edition_degree" => null "field_cover_description" => "лупа, парень едет на велосипеде" "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебное издание" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "459" "field_priority" => null "field_default_folder" => "/fizika_10/gelfgat-z/" "field_isbn" => "978-5-89237-252-7" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1250 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1251 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1252 …2} "field_url" => "/10-klass/fizika/gelfgat-zadachnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => 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#attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1316 #items: array:2 [ 0 => App\Models\Element {#1301 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1494027 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1302 #items: array:1 [ 0 => App\Models\Edition {#1300 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1303 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "stas" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1305 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1306 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1307 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => 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"*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319526" "type" => "task" ] "text" => "<p><strong>O-49.</strong> На два быстро вращающихся одинаковых валика положили горизонтально доску массой $\text{m}$ (см. рисунок). Расстояние между осями валиков $\text{L}$, коэффициент трения между доской и валиками $\mu$. Как будет двигаться доска?</p><p>Как изменится ответ, если оба валика изменят направление вращения?</p><p>☑ Доска будет совершать гармонические колебания с периодом $T = 2\pi\sqrt{L/(2\mu g)}$. Если валики изменят направление вращения, доска будет сброшена с них.</p><p><strong>Решение.</strong> В условии говорится о быстром вращении валиков; значит, между ними и доской действуют силы трения скольжения (см. рисунок).</p><p>Следовательно, $ma_x = F_{тр1} - F_{тр2} = \mu(N_1 - N_2)$, где $N_1$ и $N_2$ — силы давления доски на первый и второй валики. В положении равновесия центр доски C расположен на равных расстояниях от обоих валиков. Если доска сместилась от положения равновесия на расстояние $\text{x}$, то $\frac{N_1}{N_2} = \frac{0,5L - x}{0,5L + x}$.</p><p>Учитывая, что $N_1 + N_2 = mg$, получаем: $N_1 = mg(1/2 - x/L)$ и $N_2 = mg(1/2 + x/L)$.</p><p>Уравнение движения принимает вид $a_x = -(2\mu g/L)x$. Это уравнение гармонических колебаний с циклической частотой $\omega_0 = \sqrt{2\mu g/L}$ и периодом $T = 2\pi\sqrt{L/(2\mu g)}$.</p><p>Если валики изменят направление вращения, то обе силы трения будут направлены «наружу». Смещение доски от положения равновесия увеличит на этот раз ту силу трения, которая стремится увеличить смещение. Следовательно, равновесие в этом случае неустойчиво: доска будет сброшена с валиков.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "49-1.jpg" "alt" => null "width" => "1369" "height" => 311 "path" => "/media/fizika_10/gelfgat-z/0-olimp/49-1.webp?ts=1756251198" ] 1 => array:5 [ "name" => "49-2.jpg" "alt" => null "width" => "1343" "height" => 2144 "path" => "/media/fizika_10/gelfgat-z/0-olimp/49-2.webp?ts=1756251198" ] ] ] #original: array:7 [ "id" => 1494027 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1302} "task" => array:2 [ "refs" => "1319526" "type" => "task" ] "text" => "<p><strong>O-49.</strong> На два быстро вращающихся одинаковых валика положили горизонтально доску массой $\text{m}$ (см. рисунок). Расстояние между осями валиков $\text{L}$, коэффициент трения между доской и валиками $\mu$. Как будет двигаться доска?</p><p>Как изменится ответ, если оба валика изменят направление вращения?</p><p>☑ Доска будет совершать гармонические колебания с периодом $T = 2\pi\sqrt{L/(2\mu g)}$. Если валики изменят направление вращения, доска будет сброшена с них.</p><p><strong>Решение.</strong> В условии говорится о быстром вращении валиков; значит, между ними и доской действуют силы трения скольжения (см. рисунок).</p><p>Следовательно, $ma_x = F_{тр1} - F_{тр2} = \mu(N_1 - N_2)$, где $N_1$ и $N_2$ — силы давления доски на первый и второй валики. В положении равновесия центр доски C расположен на равных расстояниях от обоих валиков. Если доска сместилась от положения равновесия на расстояние $\text{x}$, то $\frac{N_1}{N_2} = \frac{0,5L - x}{0,5L + x}$.</p><p>Учитывая, что $N_1 + N_2 = mg$, получаем: $N_1 = mg(1/2 - x/L)$ и $N_2 = mg(1/2 + x/L)$.</p><p>Уравнение движения принимает вид $a_x = -(2\mu g/L)x$. Это уравнение гармонических колебаний с циклической частотой $\omega_0 = \sqrt{2\mu g/L}$ и периодом $T = 2\pi\sqrt{L/(2\mu g)}$.</p><p>Если валики изменят направление вращения, то обе силы трения будут направлены «наружу». Смещение доски от положения равновесия увеличит на этот раз ту силу трения, которая стремится увеличить смещение. Следовательно, равновесие в этом случае неустойчиво: доска будет сброшена с валиков.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "49-1.jpg" "alt" => null "width" => "1369" "height" => 311 "path" => "/media/fizika_10/gelfgat-z/0-olimp/49-1.webp?ts=1756251198" ] 1 => array:5 [ "name" => "49-2.jpg" "alt" => null "width" => "1343" "height" => 2144 "path" => "/media/fizika_10/gelfgat-z/0-olimp/49-2.webp?ts=1756251198" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1308 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1691584 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1309 #items: array:1 [ 0 => App\Models\Edition {#1310 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1311 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1313 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1314 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1315 …2} "field_content_source" => null ] #original: array:21 [ "id" => 6138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1311 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" …9 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1319526" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>масса доски: $m$</p><p>расстояние между осями валиков: $L$</p><p>коэффициент трения скольжения: $μ$</p><p>ускорение свободного падения: $g$</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Как будет двигаться доска.</p><p><strong>2.</strong> Как изменится характер движения, если валики изменят направление вращения.</p><p><strong>Решение:</strong></p><p><strong>Как будет двигаться доска?</strong></p><p>По умолчанию будем считать, что валики вращаются "внутрь", то есть их верхние поверхности движутся навстречу друг другу. Введем горизонтальную ось $Ox$, начало которой ($x=0$) находится ровно посередине между валиками.</p><p>Рассмотрим доску, смещенную из положения равновесия вправо на расстояние $x$. На доску действуют вертикальные силы: сила тяжести $mg$ и силы нормальной реакции опор $N_1$ (левый валик) и $N_2$ (правый валик). Из условия равновесия по вертикали:</p><p>$N_1 + N_2 = mg$</p><p>Для нахождения $N_1$ и $N_2$ по отдельности запишем уравнение моментов относительно центра масс доски C (который находится в точке $x$). Расстояние от центра масс до левого валика равно $L/2 + x$, а до правого $L/2 - x$.</p><p>$N_1(L/2 + x) - N_2(L/2 - x) = 0$</p><p>Решая эту систему уравнений, получаем:</p><p>$N_1 = mg(\frac{1}{2} - \frac{x}{L})$</p><p>$N_2 = mg(\frac{1}{2} + \frac{x}{L})$</p><p>При вращении валиков "внутрь" сила трения от левого валика $F_{тр1}$ направлена вправо, а от правого валика $F_{тр2}$ — влево. Результирующая сила, действующая на доску в горизонтальном направлении, равна:</p><p>$F_x = F_{тр1} - F_{тр2} = μN_1 - μN_2$</p><p>Подставим выражения для $N_1$ и $N_2$:</p><p>$F_x = μ[mg(\frac{1}{2} - \frac{x}{L}) - mg(\frac{1}{2} + \frac{x}{L})] = μmg(-\frac{2x}{L}) = -(\frac{2μmg}{L})x$</p><p>Согласно второму закону Ньютона, $F_x = ma_x$.</p><p>$ma_x = -(\frac{2μmg}{L})x$</p><p>Сокращая массу $m$, получаем уравнение для ускорения:</p><p>$a_x = -(\frac{2μg}{L})x$</p><p>Это уравнение описывает гармонические колебания вида $a_x = -ω^2x$. Отсюда можно найти циклическую частоту колебаний $ω$:</p><p>$ω^2 = \frac{2μg}{L} \implies ω = \sqrt{\frac{2μg}{L}}$</p><p>Период колебаний $T$ связан с циклической частотой как $T = 2π/ω$.</p><p>$T = 2π\sqrt{\frac{L}{2μg}}$</p><p>Таким образом, доска будет совершать незатухающие гармонические колебания около положения равновесия.</p><p><strong>Ответ:</strong> Доска будет совершать гармонические колебания с периодом $T = 2π\sqrt{\frac{L}{2μg}}$.</p><p><strong>Как изменится ответ, если оба валика изменят направление вращения?</strong></p><p>Если валики начнут вращаться "наружу", то направления сил трения изменятся на противоположные. Сила трения от левого валика $F_{тр1}$ будет направлена влево, а от правого $F_{тр2}$ — вправо.</p><p>Снова сместим доску вправо на расстояние $x$. Результирующая сила будет равна:</p><p>$F_x = F_{тр2} - F_{тр1} = μN_2 - μN_1$</p><p>Подставим выражения для $N_1$ и $N_2$:</p><p>$F_x = μ[mg(\frac{1}{2} + \frac{x}{L}) - mg(\frac{1}{2} - \frac{x}{L})] = μmg(\frac{2x}{L}) = +(\frac{2μmg}{L})x$</p><p>Уравнение движения примет вид:</p><p>$a_x = +(\frac{2μg}{L})x$</p><p>В этом случае результирующая сила направлена в ту же сторону, что и смещение. Такая сила не возвращает тело в положение равновесия, а наоборот, увеличивает его смещение. Положение равновесия $x=0$ становится неустойчивым. Любое случайное смещение будет нарастать, и доска будет сброшена с валиков.</p><p><strong>Ответ:</strong> Положение равновесия станет неустойчивым. При малейшем смещении от центра доска начнет ускоренно двигаться в сторону смещения и в итоге будет сброшена с валиков.</p>" ] #original: array:6 [ "id" => 1691584 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1309} "task" => array:2 [ "refs" => "1319526" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>масса доски: $m$</p><p>расстояние между осями валиков: $L$</p><p>коэффициент трения скольжения: $μ$</p><p>ускорение свободного падения: $g$</p><p><strong>Найти:</strong></p><p><strong>1.</strong> Как будет двигаться доска.</p><p><strong>2.</strong> Как изменится характер движения, если валики изменят направление вращения.</p><p><strong>Решение:</strong></p><p><strong>Как будет двигаться доска?</strong></p><p>По умолчанию будем считать, что валики вращаются "внутрь", то есть их верхние поверхности движутся навстречу друг другу. Введем горизонтальную ось $Ox$, начало которой ($x=0$) находится ровно посередине между валиками.</p><p>Рассмотрим доску, смещенную из положения равновесия вправо на расстояние $x$. На доску действуют вертикальные силы: сила тяжести $mg$ и силы нормальной реакции опор $N_1$ (левый валик) и $N_2$ (правый валик). Из условия равновесия по вертикали:</p><p>$N_1 + N_2 = mg$</p><p>Для нахождения $N_1$ и $N_2$ по отдельности запишем уравнение моментов относительно центра масс доски C (который находится в точке $x$). Расстояние от центра масс до левого валика равно $L/2 + x$, а до правого $L/2 - x$.</p><p>$N_1(L/2 + x) - N_2(L/2 - x) = 0$</p><p>Решая эту систему уравнений, получаем:</p><p>$N_1 = mg(\frac{1}{2} - \frac{x}{L})$</p><p>$N_2 = mg(\frac{1}{2} + \frac{x}{L})$</p><p>При вращении валиков "внутрь" сила трения от левого валика $F_{тр1}$ направлена вправо, а от правого валика $F_{тр2}$ — влево. Результирующая сила, действующая на доску в горизонтальном направлении, равна:</p><p>$F_x = F_{тр1} - F_{тр2} = μN_1 - μN_2$</p><p>Подставим выражения для $N_1$ и $N_2$:</p><p>$F_x = μ[mg(\frac{1}{2} - \frac{x}{L}) - mg(\frac{1}{2} + \frac{x}{L})] = μmg(-\frac{2x}{L}) = -(\frac{2μmg}{L})x$</p><p>Согласно второму закону Ньютона, $F_x = ma_x$.</p><p>$ma_x = -(\frac{2μmg}{L})x$</p><p>Сокращая массу $m$, получаем уравнение для ускорения:</p><p>$a_x = -(\frac{2μg}{L})x$</p><p>Это уравнение описывает гармонические колебания вида $a_x = -ω^2x$. Отсюда можно найти циклическую частоту колебаний $ω$:</p><p>$ω^2 = \frac{2μg}{L} \implies ω = \sqrt{\frac{2μg}{L}}$</p><p>Период колебаний $T$ связан с циклической частотой как $T = 2π/ω$.</p><p>$T = 2π\sqrt{\frac{L}{2μg}}$</p><p>Таким образом, доска будет совершать незатухающие гармонические колебания около положения равновесия.</p><p><strong>Ответ:</strong> Доска будет совершать гармонические колебания с периодом $T = 2π\sqrt{\frac{L}{2μg}}$.</p><p><strong>Как изменится ответ, если оба валика изменят направление вращения?</strong></p><p>Если валики начнут вращаться "наружу", то направления сил трения изменятся на противоположные. Сила трения от левого валика $F_{тр1}$ будет направлена влево, а от правого $F_{тр2}$ — вправо.</p><p>Снова сместим доску вправо на расстояние $x$. Результирующая сила будет равна:</p><p>$F_x = F_{тр2} - F_{тр1} = μN_2 - μN_1$</p><p>Подставим выражения для $N_1$ и $N_2$:</p><p>$F_x = μ[mg(\frac{1}{2} + \frac{x}{L}) - mg(\frac{1}{2} - \frac{x}{L})] = μmg(\frac{2x}{L}) = +(\frac{2μmg}{L})x$</p><p>Уравнение движения примет вид:</p><p>$a_x = +(\frac{2μg}{L})x$</p><p>В этом случае результирующая сила направлена в ту же сторону, что и смещение. Такая сила не возвращает тело в положение равновесия, а наоборот, увеличивает его смещение. Положение равновесия $x=0$ становится неустойчивым. Любое случайное смещение будет нарастать, и доска будет сброшена с валиков.</p><p><strong>Ответ:</strong> Положение равновесия станет неустойчивым. При малейшем смещении от центра доска начнет ускоренно двигаться в сторону смещения и в итоге будет сброшена с валиков.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1319527" "type" => "task" ] "previous" => array:2 [ "refs" => "1319525" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1350 #items: array:1 [ 0 => App\Models\Book {#1318 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 5400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1319 #items: array:1 [ 0 => App\Models\Term {#1317 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1320 #items: array:2 [ 0 => App\Models\Term {#1321 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1322 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] 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#withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1327 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] 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false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1332 #items: array:1 [ 0 => App\Models\Term {#1333 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1334 #items: array:1 [ 0 => App\Models\Term {#1335 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1336 #items: 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№49 (с. 91)
Условие. №49 (с. 91)
скриншот условия
O-49. На два быстро вращающихся одинаковых валика положили горизонтально доску массой $\text{m}$ (см. рисунок). Расстояние между осями валиков $\text{L}$, коэффициент трения между доской и валиками $\mu$. Как будет двигаться доска?
Как изменится ответ, если оба валика изменят направление вращения?
☑ Доска будет совершать гармонические колебания с периодом $T = 2\pi\sqrt{L/(2\mu g)}$. Если валики изменят направление вращения, доска будет сброшена с них.
Решение. В условии говорится о быстром вращении валиков; значит, между ними и доской действуют силы трения скольжения (см. рисунок).
Следовательно, $ma_x = F_{тр1} - F_{тр2} = \mu(N_1 - N_2)$, где $N_1$ и $N_2$ — силы давления доски на первый и второй валики. В положении равновесия центр доски C расположен на равных расстояниях от обоих валиков. Если доска сместилась от положения равновесия на расстояние $\text{x}$, то $\frac{N_1}{N_2} = \frac{0,5L - x}{0,5L + x}$.
Учитывая, что $N_1 + N_2 = mg$, получаем: $N_1 = mg(1/2 - x/L)$ и $N_2 = mg(1/2 + x/L)$.
Уравнение движения принимает вид $a_x = -(2\mu g/L)x$. Это уравнение гармонических колебаний с циклической частотой $\omega_0 = \sqrt{2\mu g/L}$ и периодом $T = 2\pi\sqrt{L/(2\mu g)}$.
Если валики изменят направление вращения, то обе силы трения будут направлены «наружу». Смещение доски от положения равновесия увеличит на этот раз ту силу трения, которая стремится увеличить смещение. Следовательно, равновесие в этом случае неустойчиво: доска будет сброшена с валиков.
Решение. №49 (с. 91)
Дано:
масса доски: $m$
расстояние между осями валиков: $L$
коэффициент трения скольжения: $μ$
ускорение свободного падения: $g$
Найти:
1. Как будет двигаться доска.
2. Как изменится характер движения, если валики изменят направление вращения.
Решение:
Как будет двигаться доска?
По умолчанию будем считать, что валики вращаются "внутрь", то есть их верхние поверхности движутся навстречу друг другу. Введем горизонтальную ось $Ox$, начало которой ($x=0$) находится ровно посередине между валиками.
Рассмотрим доску, смещенную из положения равновесия вправо на расстояние $x$. На доску действуют вертикальные силы: сила тяжести $mg$ и силы нормальной реакции опор $N_1$ (левый валик) и $N_2$ (правый валик). Из условия равновесия по вертикали:
$N_1 + N_2 = mg$
Для нахождения $N_1$ и $N_2$ по отдельности запишем уравнение моментов относительно центра масс доски C (который находится в точке $x$). Расстояние от центра масс до левого валика равно $L/2 + x$, а до правого $L/2 - x$.
$N_1(L/2 + x) - N_2(L/2 - x) = 0$
Решая эту систему уравнений, получаем:
$N_1 = mg(\frac{1}{2} - \frac{x}{L})$
$N_2 = mg(\frac{1}{2} + \frac{x}{L})$
При вращении валиков "внутрь" сила трения от левого валика $F_{тр1}$ направлена вправо, а от правого валика $F_{тр2}$ — влево. Результирующая сила, действующая на доску в горизонтальном направлении, равна:
$F_x = F_{тр1} - F_{тр2} = μN_1 - μN_2$
Подставим выражения для $N_1$ и $N_2$:
$F_x = μ[mg(\frac{1}{2} - \frac{x}{L}) - mg(\frac{1}{2} + \frac{x}{L})] = μmg(-\frac{2x}{L}) = -(\frac{2μmg}{L})x$
Согласно второму закону Ньютона, $F_x = ma_x$.
$ma_x = -(\frac{2μmg}{L})x$
Сокращая массу $m$, получаем уравнение для ускорения:
$a_x = -(\frac{2μg}{L})x$
Это уравнение описывает гармонические колебания вида $a_x = -ω^2x$. Отсюда можно найти циклическую частоту колебаний $ω$:
$ω^2 = \frac{2μg}{L} \implies ω = \sqrt{\frac{2μg}{L}}$
Период колебаний $T$ связан с циклической частотой как $T = 2π/ω$.
$T = 2π\sqrt{\frac{L}{2μg}}$
Таким образом, доска будет совершать незатухающие гармонические колебания около положения равновесия.
Ответ: Доска будет совершать гармонические колебания с периодом $T = 2π\sqrt{\frac{L}{2μg}}$.
Как изменится ответ, если оба валика изменят направление вращения?
Если валики начнут вращаться "наружу", то направления сил трения изменятся на противоположные. Сила трения от левого валика $F_{тр1}$ будет направлена влево, а от правого $F_{тр2}$ — вправо.
Снова сместим доску вправо на расстояние $x$. Результирующая сила будет равна:
$F_x = F_{тр2} - F_{тр1} = μN_2 - μN_1$
Подставим выражения для $N_1$ и $N_2$:
$F_x = μ[mg(\frac{1}{2} + \frac{x}{L}) - mg(\frac{1}{2} - \frac{x}{L})] = μmg(\frac{2x}{L}) = +(\frac{2μmg}{L})x$
Уравнение движения примет вид:
$a_x = +(\frac{2μg}{L})x$
В этом случае результирующая сила направлена в ту же сторону, что и смещение. Такая сила не возвращает тело в положение равновесия, а наоборот, увеличивает его смещение. Положение равновесия $x=0$ становится неустойчивым. Любое случайное смещение будет нарастать, и доска будет сброшена с валиков.
Ответ: Положение равновесия станет неустойчивым. При малейшем смещении от центра доска начнет ускоренно двигаться в сторону смещения и в итоге будет сброшена с валиков.
Другие задания:
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